Energy conservation on Cosmological scales

[Almighty BOB]

I'm curious to know whether anyone with good maths has anything to say about Dr Philip Gibbs' covariant formula for conserved currents of energy, momentum and angular- momentum derived from a general form of Noether’s theorem? I'm not a pro mathematician, but it looks relatively robust to me... I asked this before, but the post didn't seem to 'take'. Apologies if this is double posting - Bob edit - link. https://arxiv.org/pdf/gr-qc/9701028.pdf

 

[bhobba]

The following may be of interest in relation to this issue:
http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

Thanks
Bill

 

[Almighty BOB]

Thanks bhobba - have already had a look at that. Dr Gibbs' piece, however, claims to have done away with the need for pseudotensors; In other words he says he has shown mathematically that energy is conserved regardless of the size of the bit of spacetime you're looking at, and regardless of the reference frame. My math isn't good enough to tell if he's right or not. I do know there are people on here with top flight maths - I was just wondering if any of them could find a problem in his working, or whether he's actually answered a very big question...

 

[bhobba]

What it says in the abstract is he has found a generalization of Noether that allows a definition of energy in GR. This is exactly what John says - one can define energy in such a way its conserved, but it's not unique. The paper seems OK - but taking into account what John said - what has been done - just another definition that is conserved.

Thanks
Bill

 

[bhobba]

The following deeper look at the issue may help:
https://arxiv.org/abs/gr-qc/0403107
'A covariant generalization of the first Noether theorem for LFS is obtained.'

I seem to recall these generalizations are not unique - so its the same issue John says:
'In general, it depends on what you mean by "energy", and what you mean by "conserved".'

Looking at the paper I think its likely true but as the link I gave says its been generalized to rectify the issue before - I doubt its unique because of what John says.

Thanks
Bill

 

[Almighty BOB]

The following deeper look at the issue may help:
https://arxiv.org/abs/gr-qc/0403107
'A covariant generalization of the first Noether theorem for LFS is obtained.'

I seem to recall these generalizations are not unique - so its the same issue John says:
'In general, it depends on what you mean by "energy", and what you mean by "conserved".'

Looking at the paper I think its likely true but as the link I gave says its been generalized to rectify the issue before - I doubt its unique because of what John says.

Thanks
Bill

Thanks Bill. Do you mind if I pass it on to Dr Gibbs? Maybe he would like to sign up to the forum and put his side...

 

[bhobba]

Thanks Bill. Do you mind if I pass it on to Dr Gibbs? Maybe he would like to sign up to the forum and put his side...

Sure.

It's an old well known issue. Dr Bob is probably correct - I just suspect its not unique. I used to really be into GR and would have in the past been able to go deeper into the technicalities than I can these days. But we have people here who are real experts.

Thanks
Bill

 

[bhobba]

Hi Again

Please mention to Dr Bob I, and many on this forum agree very strongly with what is said in that paper:
The correct answer to the question “What is energy?” is that the correct energy is that which is given by Noether’s theorem.

Its just the second thing he says rears its ugly head:
This is still not unique since the Lagrangian is only unique up to a term which is a perfect divergence.

Thanks
Bill

 

[Almighty BOB]

Ho Again

Please mention to Dr Bob I, and many on this forum agree very strongly with what is said in that paper:
The correct answer to the question “What is energy?” is that the correct energy is that which is given by Noether’s theorem.

Its just the second thing he says rears its ugly head:
This is still not unique since the Lagrangian is only unique up to a term which is a perfect divergence.

Thanks
Bill

That's fascinating - ! should have paid more attention in math class! Dang!

 

[Philip Gibbs]

Someone asked me to comment about my work.

It does not matter that energy is not unique. It depends on reference frame and choice of Lagrangian. This is common in physics. Many things are relative like this. Energy is relative to the reference frame even in Newtonian mechanics. In General Relativity there is a broader choice of reference frames that are equally valid and consequently the quantity that described energy is also less unique. So long as you stick to one reference frame and one choice of Langrangian you can calculate consistently.

 

[bhobba]

That's fascinating - ! should have paid more attention in math class! Dang!

Noether is one hell of a beautiful bit of math - as well as her great courage being a female mathematician during her time:
https://www.nytimes.com/2012/03/27/...icant-mathematician-youve-never-heard-of.html

Neither Hilbert or Einstein could solve it - but Hilbert said he knew only one other who could do it - Noether - and she did. Of course she is one of the greatest female mathematicians that ever lived - maybe even the greatest and in many peoples list of greatest mathematicians all time. Inspiring.

Thanks
Bill

 

[bhobba]

Someone asked me to comment about my work. It does not matter that energy is not unique. It depends on reference frame and choice of Lagrangian. This is common in physics. Many things are relative like this. Energy is relative to the reference frame even in Newtonian mechanics. In General Relativity there is a broader choice of reference frames that are equally valid and consequently the quantity that described energy is also less unique. So long as you stick to one reference frame and one choice of Langrangian you can calculate consistently.

Yes if the uniqueness issue is even a problem is sometimes discussed here. I am with you - its irrelevant really - it just has to be consistent.

Thanks
Bill

 

[Almighty BOB]

So long as you stick to one reference frame and one choice of Langrangian you can calculate consistently.

The question then would appear to be why other cosmologists don't seem prepared to accept this? Is it simply a case of arguing over definitions? Or have I (as usual) missed something important?

 

[bhobba]

The question then would appear to be why other cosmologists don't seem prepared to accept this? Is it simply a case of arguing over definitions? Or have I (as usual) missed something important?

if you hang around here you find all sorts of views on all sorts of things. If it can't be disproved by experiment and is reasonable then its open for debate. I just happen to agree with Dr Bob on this - you may find others that disagree,

This occurs not so much in GR (but does occur), but infests QM a lot.

Always keep in mind Feynman:


Thanks
Bill

 

[Almighty BOB]

if you hang around here you find all sorts of views on all sorts of things. If it can't be disproved by experiment and is reasonable then its open for debate. I just happen to agree with Dr Bob on this - you may find others that disagree,

This occurs not do much in GR (but does occur), but infests QM a lot.

Always keep in mind Feynman:


Thanks
Bill

So you don't think a rigorous mathematical proof answers the question? Or you're saying there's simply no empirical evidence one way or the other? (asking for a friend :-) )

 

[bhobba]

So you don't think a rigorous mathematical proof answers the question? Or you're saying there's simply no empirical evidence one way or the other? (asking for a friend :-) )

I am saying read what John Baez said. There are a number of Pseudo Tensor definitions. Now I agree the answer is Noether - but if it agrees with experiment then how do you prove or disprove something. Dr Bob's method is very elegant and I am convinced - but what was it Einstein said - elegance is for Taylors.

Thanks
Bill

 

[Almighty BOB]

elegance is for Taylors.

Bill

You should tell my tailor that. Two left hands. At least he's cheap. Still reading Baez, but what I'm currently getting is that there is no problem with the derivation of conserved energy with respect to any given choice of time translation. It's conserved from any specific frame of reference, in short. So it looks like a bit of a non-argument to me...

 

[bhobba]

So it looks like a bit of a non-argument to me...

What can I say - I am with you but you soon find those that disagree. That's one reason we have mentors - to ensure it doesn't get out of hand.

Thanks
Bill

 

[Philip Gibbs]

Two pulsars radiate energy as gravitational waves and come closer together.
Two black holes merge and the gravitational waves are detected on Earth.
Those are the experiments. The problem is more theoretical. People don't accept the formulation because energy takes the form of a kind of group representation that they are not familiar with. I.e. it is from a dual representation of the diffeomorphism group.
The answer I get reduces to the Komar Superpotential which has been known about and used for decades. All I did was relate it to Noether.
But even Komar did not quite recognise it as a general solution to the problem of energy conservation. They expect something like a 4-vector but the answer is something else they don't recognise.

 

[Almighty BOB]

What can I say - I am with you but you soon find those that disagree. That's one reason we have mentors - to ensure it doesn't get out of hand.

Thanks
Bill

Thanks Bill, I've read the Baez page now - yeah, what you just wrote makes perfect sense :-) Catch you later!

 

[Philip Gibbs]

When Dirac started to use delta functions mathematicians said it was wrong, until they realized that he was working with the dual vector space to the space of continuous functions. Energy conservation in GR uses the same trick. Conserved quantities come from the space dual to fields in space-time. That makes them look odd but once understood in those terms everything is fine. In special relativity nobody notices because the relevant vector fields are finite dimensional and vector spaces are isomorphic to their duals. In GR the group of transformations is infinite dimensional so the answer is harder to understand, but it works just fine.

 

[bhobba]

Two pulsars radiate energy as gravitational waves and come closer together.
Two black holes merge and the gravitational waves are detected on Earth.Those are the experiments. The problem is more theoretical. People don't accept the formulation because energy takes the form of a kind of group representation that they are not familiar with. I.e. it is from a dual representation of the diffeomorphism group. The answer I get reduces to the Komar Superpotential which has been known about and used for decades. All I did was relate it to Noether.But even Komar did not quite recognise it as a general solution to the problem of energy conservation. They expect something like a 4-vector but the answer is something else they don't recognise.

We have some people with enough background to fully understand what you said. I can send them a note if you want to discuss it.

But IMHO Noether defines energy - the other stuff to me is irrelevant - I mean group representation they are not familiar with - who cares. I am not familiar with a lot of stuff such as exactly what is going on with 1 + 2 + 3 +4 ... = -1/12. I suspect the answer has something to do with re-normalization but it is taking me into areas I am not familiar with. Such things are hard at first but seem easier with acquaintance and time. And there are things I never have understood doesn't matter how hard I try such as the conventional method of re-normalization - but I do understand BPH re-normalization which is supposed to be equivalent. What can you do - just keep plugging away with an open mind.

Thanks
Bill

 

[bhobba]

When Dirac started to use delta functions mathematicians said it was wrong, until they realized that he was working with the dual vector space to the space of continuous functions.

I know that one only too well - I have to explain Rigged Hibert Spaces to many who actually think about it - eg e^ipx is not square integrable. And a rigorous treatment leading to the generalized eigenvalue theorem is quite advanced - so advanced the proof given in Gelfland's tome is wrong - but I have another proof valid for cases in QM I give out.

Von-Neumann was scathing - largely because I think he and Hilbert could not figure it out. But Grothendieck, Gelfland and Schwartz did and as a side benefit made Fourier Analysis a lot easier.

It just takes time.

Thanks
Bill

 

[Almighty BOB]

Is Professor Neumaier still active on these forums? I'm sure he could critique Gibbs' work. The professor is a maths genius, from what I've heard.

 

[bhobba]

Is Professor Neumaier still active on these forums? I'm sure he could critique Gibbs' work. The professor is a maths genius, from what I've heard.

Most definitely. I don't think Dr Neumaier is just a math's genius - he is a genius period.

You can drop him a note about it if you like.

Thanks
Bill